Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129125 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1776 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C2 | #10821 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
459 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,17)(4,23,27,46)(5,13)(6,32)(7,12)(8,35)(9,16)(10,19)(11,26,42,34)(14,31)(15,30)(18,36,52,62)(20,38)(21,45)(22,44)(24,56,57,40)(25,43,49,60)(28,41,59,55)(29,39)(33,48)(37,54)(47,51,64,53)(50,61,58,63);(1,3,7,20,8,21,31,19,9,22,6,33,30,54,5,29)(2,10,14,37,15,38,12,44,16,39,13,45,35,48,32,17)(4,24,26,58,28,52,25,51,27,57,34,50,59,18,49,53)(11,40,23,47,43,62,55,61,42,56,46,64,60,36,41,63);(1,4,5,25,30,59,6,26,9,27,31,49,8,28,7,34)(2,11,12,41,35,60,14,46,16,42,32,55,15,43,13,23)(3,18,19,50,54,57,20,51,22,52,29,58,21,24,33,53)(10,36,17,47,48,56,45,63,39,62,44,64,38,40,37,61)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307209, #72307247, #72307268, … |
| maximal normal subgroup of | #72307209, #72307247, #72307268, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54130);